Breaking the Linear-Memory Barrier in $$\mathsf {MPC}$$ : Fast $$\mathsf {MIS}$$ on Trees with Strongly Sublinear Memory

2019 ◽  
pp. 124-138 ◽  
Author(s):  
Sebastian Brandt ◽  
Manuela Fischer ◽  
Jara Uitto
Keyword(s):  
Strongly Sublinear

scholarly journals On the oscillation of a nonlinear two-dimensional difference systems

2001 ◽  
Vol 32 (3) ◽  
pp. 201-209 ◽  
Author(s):  
E. Thandapani ◽  
B. Ponnammal
Keyword(s):  
Asymptotic Behavior ◽  
Sufficient Conditions ◽  
Real Sequence ◽  
Two Dimensional ◽  
Difference System ◽  
Nonoscillatory Solutions ◽  
Difference Systems ◽  
All Solutions ◽  
Necessary And Sufficient ◽  
Strongly Sublinear

The authors consider the two-dimensional difference system$$ \Delta x_n = b_n g (y_n) $$ $$ \Delta y_n = -f(n, x_{n+1}) $$where $ n \in N(n_0) = \{ n_0, n_0+1, \ldots \} $, $ n_0 $ a nonnegative integer; $ \{ b_n \} $ is a real sequence, $ f: N(n_0) \times {\rm R} \to {\rm R} $ is continuous with $ u f(n,u) > 0 $ for all $ u \ne 0 $. Necessary and sufficient conditions for the existence of nonoscillatory solutions with a specified asymptotic behavior are given. Also sufficient conditions for all solutions to be oscillatory are obtained if $ f $ is either strongly sublinear or strongly superlinear. Examples of their results are also inserted.

scholarly journals Strongly Sublinear Separators and Polynomial Expansion

2016 ◽  
Vol 30 (2) ◽  
pp. 1095-1101 ◽  
Author(s):  
Zdeněk Dvořák ◽  
Sergey Norin
Keyword(s):  
Polynomial Expansion ◽  
Strongly Sublinear

On the oscillation of fourth order strongly superlinear and strongly sublinear dynamic equations

2013 ◽  
Vol 44 (1-2) ◽  
pp. 119-132 ◽  
Author(s):  
Said R. Grace ◽  
Shurong Sun ◽  
Yizhuo Wang
Keyword(s):  
Fourth Order ◽  
Dynamic Equations ◽  
Strongly Sublinear

Oscillation of second-order strongly superlinear and strongly sublinear dynamic equations

2009 ◽  
Vol 14 (8) ◽  
pp. 3463-3471 ◽  
Author(s):  
Said R. Grace ◽  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Donal O’Regan
Keyword(s):  
Second Order ◽  
Dynamic Equations ◽  
Strongly Sublinear

scholarly journals Smaller Extended Formulations for Spanning Tree Polytopes in Minor-closed Classes and Beyond

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Manuel Aprile ◽  
Samuel Fiorini ◽  
Tony Huynh ◽  
Gwenaël Joret ◽  
David R. Wood
Keyword(s):  
Spanning Tree ◽  
Short Proof ◽  
Induced Subgraphs ◽  
Direct Construction ◽  
Extension Complexity ◽  
Graph Classes ◽  
Closed Class ◽  
Vertex Graph ◽  
Graph Class ◽  
Strongly Sublinear

Let $G$ be a connected $n$-vertex graph in a proper minor-closed class $\mathcal G$. We prove that the extension complexity of the spanning tree polytope of $G$ is $O(n^{3/2})$. This improves on the $O(n^2)$ bounds following from the work of Wong (1980) and Martin (1991). It also extends a result of Fiorini, Huynh, Joret, and Pashkovich (2017), who obtained a $O(n^{3/2})$ bound for graphs embedded in a fixed surface. Our proof works more generally for all graph classes admitting strongly sublinear balanced separators: We prove that for every constant $\beta$ with $0<\beta<1$, if $\mathcal G$ is a graph class closed under induced subgraphs such that all $n$-vertex graphs in $\mathcal G$ have balanced separators of size $O(n^\beta)$, then the extension complexity of the spanning tree polytope of every connected $n$-vertex graph in $\mathcal{G}$ is $O(n^{1+\beta})$. We in fact give two proofs of this result, one is a direct construction of the extended formulation, the other is via communication protocols. Using the latter approach we also give a short proof of the $O(n)$ bound for planar graphs due to Williams (2002).

scholarly journals OSCILLATION OF HIGHER ORDER STRONGLY SUPERLINEAR AND STRONGLY SUBLINEAR DIFFERENCE EQUATIONS

2014 ◽  
Vol 32 (3_4) ◽  
pp. 455-464
Author(s):  
Said R. Grace ◽  
Zhenlai Han ◽  
Xinhui Li
Keyword(s):  
Difference Equations ◽  
Higher Order ◽  
Strongly Sublinear

Strongly sublinear growth of the photorefractive effect for increasing pump intensities in doped lithium-niobate crystals

2007 ◽  
Vol 101 (11) ◽  
pp. 116105 ◽  
Author(s):  
Paolo Minzioni ◽  
Ilaria Cristiani ◽  
Vittorio Degiorgio ◽  
Edvard P. Kokanyan
Keyword(s):  
Lithium Niobate ◽  
Photorefractive Effect ◽  
Strongly Sublinear ◽  
Lithium Niobate Crystals

scholarly journals Text Indexing for Regular Expression Matching

Algorithms ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 133
Author(s):  
Daniel Gibney ◽  
Sharma V. Thankachan
Keyword(s):  
Time Complexity ◽  
Regular Expression ◽  
Fundamental Problem ◽  
Matrix Multiplication ◽  
Query Time ◽  
Algorithmic Problem ◽  
Time And Space ◽  
Matrix Vector Multiplication ◽  
Deterministic Solution ◽  
Strongly Sublinear

Finding substrings of a text T that match a regular expression p is a fundamental problem. Despite being the subject of extensive research, no solution with a time complexity significantly better than O(|T||p|) has been found. Backurs and Indyk in FOCS 2016 established conditional lower bounds for the algorithmic problem based on the Strong Exponential Time Hypothesis that helps explain this difficulty. A natural question is whether we can improve the time complexity for matching the regular expression by preprocessing the text T? We show that conditioned on the Online Matrix–Vector Multiplication (OMv) conjecture, even with arbitrary polynomial preprocessing time, a regular expression query on a text cannot be answered in strongly sublinear time, i.e., O(|T|1−ε) for any ε>0. Furthermore, if we extend the OMv conjecture to a plausible conjecture regarding Boolean matrix multiplication with polynomial preprocessing time, which we call Online Matrix–Matrix Multiplication (OMM), we can strengthen this hardness result to there being no solution with a query time that is O(|T|3/2−ε). These results hold for alphabet sizes three or greater. We then provide data structures that answer queries in O(|T||p|τ) time where τ∈[1,|T|] is fixed at construction. These include a solution that works for all regular expressions with Expτ·|T| preprocessing time and space. For patterns containing only ‘concatenation’ and ‘or’ operators (the same type used in the hardness result), we provide (1) a deterministic solution which requires Expτ·|T|log2|T| preprocessing time and space, and (2) when |p|≤|T|z for z=2o(log|T|), a randomized solution with amortized query time which answers queries correctly with high probability, requiring Expτ·|T|2Ωlog|T| preprocessing time and space.

scholarly journals Information capacity of specific interactions

2016 ◽  
Vol 113 (21) ◽  
pp. 5841-5846 ◽  
Author(s):  
Miriam H. Huntley ◽  
Arvind Murugan ◽  
Michael P. Brenner
Keyword(s):  
Self Assembly ◽  
Physical Parameters ◽  
Specific Interactions ◽  
Color Coding ◽  
Chemical Mechanisms ◽  
Experimental Systems ◽  
Shape Coding ◽  
Target Binding ◽  
Strongly Sublinear ◽  
Physical And Chemical

Specific interactions are a hallmark feature of self-assembly and signal-processing systems in both synthetic and biological settings. Specificity between components may arise from a wide variety of physical and chemical mechanisms in diverse contexts, from DNA hybridization to shape-sensitive depletion interactions. Despite this diversity, all systems that rely on interaction specificity operate under the constraint that increasing the number of distinct components inevitably increases off-target binding. Here we introduce “capacity,” the maximal information encodable using specific interactions, to compare specificity across diverse experimental systems and to compute how specificity changes with physical parameters. Using this framework, we find that “shape” coding of interactions has higher capacity than chemical (“color”) coding because the strength of off-target binding is strongly sublinear in binding-site size for shapes while being linear for colors. We also find that different specificity mechanisms, such as shape and color, can be combined in a synergistic manner, giving a capacity greater than the sum of the parts.

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